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Robust Parallel Smoothing for Multigrid via Sparse Approximate Inverses

by O. Broeker and M. J. Grote and A. Reusken

(Report number 2000-13)

Abstract
Sparse approximate inverses are considered as smoothers for multigrid. They are based on the SPAI-Algorithm (Grote and Huckle, 1997), which constructs a sparse approximate inverse M of a matrix A by minimizing I -MA in the Frobenius norm. This yields a new hierarchy of smoothers: SPAI-0, SPAI-1, SPAI$(\varepsilon)$. Advantages of SPAI smoothers over classical smoothers are inherent parallelism, possible local adaptivity and improved robustness. The simplest smoother, SPAI-0, is based on a diagonal matrix M. It is shown to satisfy the smoothing property for symmetric positive definite problems. Numerical experiments show that SPAI-0 smoothing is usually preferable to damped Jacobi smoothing. In more difficult situations, where the simpler SPAI-0 and SPAI-1 smoothers are not adequate, the SPAI$(\varepsilon)$ smoother provides a natural procedure for improvement where needed. Numerical examples illustrate the usefulness of SPAI smoothing.

Keywords:

@Techreport{BGR00_272,
  author = {O. Broeker and M. J. Grote and A. Reusken},
  title = {Robust Parallel Smoothing for Multigrid via Sparse Approximate Inverses},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2000-13},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2000/2000-13.pdf },
  year = {2000}
}

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